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import OAI.Probability.InvariantIsing.Fields.FieldTerminalOrder
namespace OAI
/-! Exact single-height derivatives of the finite Gaussian field value.
An interior height changes its two adjacent covariance increments with
opposite signs. The final height includes the diagonal subtraction. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
open scoped NNReal
namespace InvariantIsing
def fieldFixedAffine (S : List (ℝ × ℝ≥0)) : List FieldAffineStep :=
S.map fun av => ⟨av.2, 0, av.1⟩
lemma fieldFixedAffine_base (S : List (ℝ × ℝ≥0))
(hS : ∀ av ∈ S, 0 < (av.2 : ℝ)) :
∀ av ∈ fieldFixedAffine S, 0 < av.base := by
intro av hav
obtain ⟨bv, hb, rfl⟩ := List.mem_map.mp hav
exact hS bv hb
lemma fieldFixedAffine_slope (S : List (ℝ × ℝ≥0)) :
∀ av ∈ fieldFixedAffine S, av.slope = 0 := by
intro av hav
obtain ⟨bv, _, rfl⟩ := List.mem_map.mp hav
rfl
lemma fieldFixedAffine_increments (S : List (ℝ × ℝ≥0)) (t : ℝ) :
fieldAffineIncrements (fieldFixedAffine S) t = S := by
simp only [fieldAffineIncrements, fieldFixedAffine, List.map_map, Function.comp_def]
have he : (fun av : ℝ × ℝ≥0 =>
(((⟨av.2, 0, av.1⟩ : FieldAffineStep).exponent),
Real.toNNReal ((⟨av.2, 0, av.1⟩ : FieldAffineStep).base +
(⟨av.2, 0, av.1⟩ : FieldAffineStep).slope * t))) = id := by
funext av
simp
rw [he, List.map_id]
/-- Actual increment list when a single nonterminal height is varied. -/
def fieldAdjacentIncrements (P S : List (ℝ × ℝ≥0)) (a b ζ η t : ℝ) :
List (ℝ × ℝ≥0) :=
P ++ (ζ, Real.toNNReal (a + t)) :: (η, Real.toNNReal (b - t)) :: S
def fieldAdjacentLevel (P S : List (ℝ × ℝ≥0)) (a b ζ η t : ℝ) :
Fin ((fieldAdjacentIncrements P S a b ζ η t).length + 1) :=
fieldAppendIndex P ((ζ, Real.toNNReal (a + t)) ::
(η, Real.toNNReal (b - t)) :: S) 1
private lemma adjacent_value (a b ζ η : ℝ) (S : List (ℝ × ℝ≥0))
(hS : ∀ av ∈ S, 0 < (av.2 : ℝ)) (t : ℝ) :
(fun z => (fieldAdjacentFamily a b ζ η (fieldFixedAffine S)
(fieldFixedAffine_base S hS) (fieldFixedAffine_slope S)).U (t, z)) =
fieldScalarValue ((ζ, Real.toNNReal (a + t)) ::
(η, Real.toNNReal (b - t)) :: S) (fun z => Real.log (Real.cosh z)) := by
change gaussianTransform (a + 1 * t) ζ
(fun y => gaussianTransform (b + -1 * t) η
(fun z => (fieldAdjacentTail a b (fieldFixedAffine S)
(fieldFixedAffine_base S hS) (fieldFixedAffine_slope S)).U (t, z)) y) = _
rw [one_mul, neg_one_mul, ← sub_eq_add_neg, field_gaussianTransform_eq_operator]
have he : (fun y => gaussianTransform (b - t) η
(fun z => (fieldAdjacentTail a b (fieldFixedAffine S)
(fieldFixedAffine_base S hS) (fieldFixedAffine_slope S)).U (t, z)) y) =
gaussianOperator η (Real.toNNReal (b - t))
(fieldScalarValue S (fun z => Real.log (Real.cosh z))) := by
rw [field_gaussianTransform_eq_operator]
rw [show (fun z => (fieldAdjacentTail a b (fieldFixedAffine S)
(fieldFixedAffine_base S hS) (fieldFixedAffine_slope S)).U (t, z)) =
(fun z => fieldAffineValue (fieldFixedAffine S) (t, z)) from
congrArg (fun f => fun z => f (t, z))
(fieldAffineFamily_value _ _ _ _),
fieldAffineValue_scalar, fieldFixedAffine_increments]
rw [he]
rfl
private lemma adjacent_inner_mean (a b η : ℝ) (S : List (ℝ × ℝ≥0))
(hS : ∀ av ∈ S, 0 < (av.2 : ℝ)) (t : ℝ) :
(fun z => (fieldAdjacentInner a b η (fieldFixedAffine S)
(fieldFixedAffine_base S hS) (fieldFixedAffine_slope S)).X (t, z)) =
fieldScalarMean ((η, Real.toNNReal (b - t)) :: S)
(fun z => Real.log (Real.cosh z)) Real.tanh := by
let av : FieldAffineStep := ⟨b, -1, η⟩
have hv : ∀ v ∈ av :: fieldFixedAffine S, ∀ q ∈ Ioo (-a) b,
0 < v.base + v.slope * q := by
intro v hv q hq
rcases List.mem_cons.mp hv with rfl | hv
· dsimp only [av]
linarith [hq.2]
· rw [fieldFixedAffine_slope S v hv, zero_mul, add_zero]
exact fieldFixedAffine_base S hS v hv
have h := fieldAffineFamily_mean (Ioo (-a) b) isOpen_Ioo
(av :: fieldFixedAffine S) hv t
change (fun z => (fieldAffineFamily (Ioo (-a) b) isOpen_Ioo
(av :: fieldFixedAffine S) hv).X (t, z)) = _
rw [h]
simp only [fieldAffineIncrements, List.map_cons]
rw [← fieldAffineIncrements, fieldFixedAffine_increments]
simp only [av, neg_one_mul, ← sub_eq_add_neg]
/-- The manuscript's finite interior-height gradient before the fixed
normalizing diagonal term. The derivative is minus half the cell width
multiplied by its actual conditional squared spin mean. -/
theorem hasDerivAt_fieldAdjacentValue
(P S : List (ℝ × ℝ≥0))
(hP : ∀ av ∈ P, 0 < (av.2 : ℝ))
(hS : ∀ av ∈ S, 0 < (av.2 : ℝ))
(a b ζ η t z : ℝ) (ht : t ∈ Ioo (-a) b) :
HasDerivAt (fun q => fieldScalarValue (fieldAdjacentIncrements P S a b ζ η q)
(fun y => Real.log (Real.cosh y)) z)
(-(η - ζ) / 2 * fieldScalarSquares (fieldAdjacentIncrements P S a b ζ η t)
(fun y => Real.log (Real.cosh y)) Real.tanh
(fieldAdjacentLevel P S a b ζ η t) z) t := by
let L := fieldFixedAffine S
let hb := fieldFixedAffine_base S hS
let hs := fieldFixedAffine_slope S
let F := fieldAdjacentTail a b L hb hs
let G := fieldAdjacentInner a b η L hb hs
let H := fieldAdjacentFamily a b ζ η L hb hs
let K := fieldConstantPrefix isOpen_Ioo H P hP
have hval (q : ℝ) : (fun y => K.U (q, y)) =
fieldScalarValue (fieldAdjacentIncrements P S a b ζ η q)
(fun y => Real.log (Real.cosh y)) := by
rw [show (fun y => K.U (q, y)) =
fieldScalarValue P (fun y => H.U (q, y)) from
fieldConstantPrefix_value isOpen_Ioo H P hP q,
adjacent_value a b ζ η S hS q]
exact (fieldScalarValue_append P _ _).symm
have htan : (fun y => H.T (t, y)) = fun y =>
-(η - ζ) / 2 * gaussianTiltAverage (a + t) ζ
(fun x => G.U (t, x)) (fun x => (G.X (t, x)) ^ 2) y := by
funext y
exact F.adjacent_tangent isOpen_Ioo a b ζ η
(fun q hq => by linarith [hq.1]) (fun q hq => by linarith [hq.2])
(p := (t, y)) ht (fun x => fieldAdjacentTail_stationary a b L hb hs (t, x))
have hsquare : gaussianTiltAverage (a + t) ζ
(fun x => G.U (t, x)) (fun x => (G.X (t, x)) ^ 2) =
fieldScalarSquares ((ζ, Real.toNNReal (a + t)) ::
(η, Real.toNNReal (b - t)) :: S)
(fun y => Real.log (Real.cosh y)) Real.tanh 1 := by
have hm : Measurable (fun x => (G.X (t, x)) ^ 2) :=
(G.mX.comp (by fun_prop)).pow_const 2
have hmG : Measurable (fun x => G.U (t, x)) := G.mU.comp (by fun_prop)
have htrans : gaussianTiltAverage (a + t) ζ
(fun x => G.U (t, x)) (fun x => (G.X (t, x)) ^ 2) =
fieldSpinTransition ζ (Real.toNNReal (a + t))
(fun x => G.U (t, x)) (fun x => (G.X (t, x)) ^ 2) := by
exact field_gaussianTiltAverage_eq_transition _ _ hmG hm
rw [htrans]
have hG : (fun x => G.U (t, x)) = fieldScalarValue
((η, Real.toNNReal (b - t)) :: S) (fun y => Real.log (Real.cosh y)) := by
change gaussianTransform (b + -1 * t) η
(fun x => F.U (t, x)) = _
rw [neg_one_mul, ← sub_eq_add_neg, field_gaussianTransform_eq_operator]
congr 1
rw [show (fun x => F.U (t, x)) = (fun x => fieldAffineValue L (t, x)) from
congrArg (fun f => fun x => f (t, x)) (fieldAffineFamily_value _ _ _ _),
fieldAffineValue_scalar, fieldFixedAffine_increments]
rfl
rw [hG]
have hGX : (fun x => G.X (t, x)) = fieldScalarMean
((η, Real.toNNReal (b - t)) :: S)
(fun y => Real.log (Real.cosh y)) Real.tanh := adjacent_inner_mean a b η S hS t
have hpow : (fun x => (G.X (t, x)) ^ 2) = fun x => (fieldScalarMean
((η, Real.toNNReal (b - t)) :: S)
(fun y => Real.log (Real.cosh y)) Real.tanh x) ^ 2 := by
funext x
rw [congrFun hGX x]
rw [hpow]
rfl
have hd := K.parameter_derivative t z ht
have hfun : (fun q => K.U (q, z)) = fun q =>
fieldScalarValue (fieldAdjacentIncrements P S a b ζ η q)
(fun y => Real.log (Real.cosh y)) z := by
funext q
exact congrFun (hval q) z
rw [hfun] at hd
convert hd using 1
rw [show K.T (t, z) = fieldScalarMean P (fun y => H.U (t, y))
(fun y => H.T (t, y)) z from
congrFun (fieldConstantPrefix_tangent isOpen_Ioo H P hP t) z,
htan, fieldScalarMean_const_mul, hsquare, adjacent_value a b ζ η S hS t]
simp only [fieldAdjacentLevel, fieldAdjacentIncrements]
rw [fieldScalarSquares_append]
/-- The terminal height changes the final variance and the normalizing
subtraction by the same amount. -/
def fieldTerminalIncrements (P : List (ℝ × ℝ≥0)) (a ζ t : ℝ) :
List (ℝ × ℝ≥0) := P ++ [(ζ, Real.toNNReal (a + t))]
def fieldTerminalLevel (P : List (ℝ × ℝ≥0)) (a ζ t : ℝ) :
Fin ((fieldTerminalIncrements P a ζ t).length + 1) :=
fieldAppendIndex P [(ζ, Real.toNNReal (a + t))] 1
theorem hasDerivAt_fieldTerminalValue
(P : List (ℝ × ℝ≥0)) (hP : ∀ av ∈ P, 0 < (av.2 : ℝ))
(a ζ d t z : ℝ) (ht : t ∈ Ioi (-a)) :
HasDerivAt (fun q => fieldScalarValue (fieldTerminalIncrements P a ζ q)
(fun y => Real.log (Real.cosh y)) z - (d + q) / 2)
(-(1 - ζ) / 2 * fieldScalarSquares (fieldTerminalIncrements P a ζ t)
(fun y => Real.log (Real.cosh y)) Real.tanh (fieldTerminalLevel P a ζ t) z) t := by
let H := fieldTerminalFamily a ζ
let K := fieldConstantPrefix isOpen_Ioi H P hP
let A : ℝ → ℝ := fun y => gaussianTiltAverage (a + t) ζ
(fun x => Real.log (Real.cosh x)) (fun x => (Real.tanh x) ^ 2) y
have hm : Measurable (fun x : ℝ => (Real.tanh x) ^ 2) := by
simp only [Real.tanh_eq]
fun_prop
have hA : Measurable A :=
measurable_gaussianTiltAverage_param (measurable_logCosh.comp measurable_snd)
(hm.comp measurable_snd) measurable_const measurable_id ζ
have hAb : ∀ y, |A y| ≤ 1 := by
intro y
apply field_gaussianTiltAverage_bound _ _ measurable_logCosh logCosh_linearGrowth hm
intro x
rw [abs_pow]
nlinarith [field_abs_tanh_le_one x, abs_nonneg (Real.tanh x)]
have hval (q : ℝ) : (fun y => K.U (q, y)) =
fieldScalarValue (fieldTerminalIncrements P a ζ q)
(fun y => Real.log (Real.cosh y)) := by
rw [show (fun y => K.U (q, y)) =
fieldScalarValue P (fun y => H.U (q, y)) from
fieldConstantPrefix_value isOpen_Ioi H P hP q,
fieldTerminalFamily_value_eq_operator]
change fieldScalarValue P (fieldScalarValue [(ζ, Real.toNNReal (a + q))]
(fun y => Real.log (Real.cosh y))) = _
exact (fieldScalarValue_append P [(ζ, Real.toNNReal (a + q))]
(fun y => Real.log (Real.cosh y))).symm
have htangent : (fun y => H.T (t, y)) = fun y => 1 / 2 + (-(1 - ζ) / 2) * A y := by
funext y
rw [fieldTerminalFamily_tangent]
dsimp only [A]
ring
have hsquare : A = fieldScalarSquares [(ζ, Real.toNNReal (a + t))]
(fun y => Real.log (Real.cosh y)) Real.tanh 1 := by
dsimp only [A]
rw [field_gaussianTiltAverage_eq_transition _ _ measurable_logCosh hm]
rfl
have hd := (K.parameter_derivative t z ht).sub
(((hasDerivAt_const t d).add (hasDerivAt_id t)).div_const 2)
change HasDerivAt (fun q => K.U (q, z) - (d + q) / 2)
(K.T (t, z) - (0 + 1) / 2) t at hd
have hfun : (fun q => K.U (q, z) - (d + q) / 2) = fun q =>
fieldScalarValue (fieldTerminalIncrements P a ζ q)
(fun y => Real.log (Real.cosh y)) z - (d + q) / 2 := by
funext q
rw [congrFun (hval q) z]
rw [hfun] at hd
convert hd using 1
rw [show K.T (t, z) = fieldScalarMean P (fun y => H.U (t, y))
(fun y => H.T (t, y)) z from
congrFun (fieldConstantPrefix_tangent isOpen_Ioi H P hP t) z,
htangent, fieldConstantPrefix_test_affine isOpen_Ioi H P hP t hA hAb,
hsquare, fieldTerminalFamily_value_eq_operator]
simp only [fieldTerminalLevel, fieldTerminalIncrements]
rw [fieldScalarSquares_append]
change _ = (1 / 2 + -(1 - ζ) / 2 * _) - (0 + 1) / 2
simp only [fieldScalarValue, List.foldr_cons, List.foldr_nil]
ring
end InvariantIsing
end
end OAI
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